Thaw puff pastry for at least 30 minutes, or until dough is soft and pliable.
Peel and dice apples, then combine them in a bowl with brown sugar, cinnamon, white sugar, and lime juice. Mix thoroughly until evenly coated.
Roll out or unfold your puff pastry. With a sharp knife, make a series of cuts one finger width apart, all the way up each side panel in a diagonal direction.
Spread your apples out on the middle panel, so that they cover it completely. Leave 1 inch clear on each end, and tuck them over to cover the ends.
Grab your bottom-most dough “string, and fold over all the way so that it covers your apples to the other side. Repeat this, alternating sides each time, so you have a criss cross pattern going up.
When you are finished, move your braid to a baking sheet lined with parchment paper. Brush on melted butter so that it covers the entire pastry.
Bake at 375 degrees for 20 minutes.
Serve as-is or drizzle with icing.